Universal Introduction to the Quantum Hall Effectijapm.org/papers/207-P0055.pdf · The quantum Hall...

Abstract—The Hall effect is the generation of a current

perpendicular to the direction of applied electric as well as

applied magnetic field in a metal or a semiconductor. It is used

to determine the concentration of electrons. The Quantum Hall

effect has been discovered by von Klitzing in Germany and by

Tsui, Stormer and Gossard in U.S.A. Robert Laughlin also in

U.S.A. explained the quantization of Hall current by using “flux

quantization” and introduced incompressibility to obtain

fractional charge. We have developed the theory of the

quantum Hall effect by using the theory of angular momentum.

Our predicted fractions are in accord with those measured. Von

Klitzing in 1985, and Tsui, Stormer and Laughlin in 1998

received the Nobel prize for their discoveries. In this lecture, we

report a review of the subject as well as emphasize our

explanation of the observed phenomena. We use spin to explain

the fractional charge and hence we discover spin-charge

coupling. We find a new way to quantize the flux and compare

our explanation with other prize winning researches.

Index Terms—Angular momentum, flux quantization, hall

effect, modified landau levels, modified lande’s g value formula.

I. INTRODUCTION

The ordinary Hall effect was discovered by Edwin Hall [1]

in 1879. In 1930, Landau showed that the orbital motion of

the electron gives magnetic susceptibility. In 1978 K. von

Klitzing and Th. Englert [2] found a plateau in the Hall effect.

In 1980 von Klitzing et al. [3] found the value of h/e2 from

the plateau in the Hall effect. In 1985, Klitzing was awarded

the Nobel prize in Physics for the discovery of quantum Hall

effect. In 1982, D. C. Tsui, H. L. Stormer and A. C. Gossard

[4] discovered the steps at fractional numbers which was

extended by Willet et al. [5]. Laughlin wrote a wave function

which gave the ideas of a fractional charge [6]. Shrivastava [7]

wrote the correct theory which agrees with the experimental

data. In 1998, Tsui, Stormer and Laughlin were awarded the

Nobel prize.

A. Applications of the Hall Effect

1) Hall probe. The detection of magnetic fields is often

done by using Hall currents.

2) Hall effect sensors. The sensor responds to changes in

the magnetic field intensity.

3) Hall effect motors and switches.

The force is F= e vxB so that the Hall voltage is, V=

IB/necd

where I is the Hall current, B is the magnetic induction, n the

electron concentration, e is the electron charge, c is the

velocity of light and d is the thickness of the sample as shown

Manuscript received February 7, 2013; revise March 22, 2013.

K. N. Shrivastava is with the University of Hyderabad, Hyderabad

500046, India (e-mail: [email protected]).

in the Fig. 1.

Fig. 1. The Hall voltage is measured orthogonal to both the applied electric as

well as the magnetic field.

B. Two Dimensional Electron Systems

A two dimensional electron system is formed in a

heterostructure which has layers of GaAs over AlGaAs. The

energy gap of GaAs increases upon Al doping. When GaAs

is doped with donors at zero temperature the Fermi level lies

higher than the bottom of the conduction band. The electrons

bound to donors move into GaAs conduction band and the

process stops when some proportion of electrons have moved.

The electrons in the inversion layer are two dimensional as

shown in Fig. 2.

The average drift velocity of the electron subjected to the

electric field is,

Vd=-eE/m (1)

where E is the electric field, m is the electron mass and is

the mean life time so that the current density is,

j=-neVd=oE (2)

where

Universal Introduction to the Quantum Hall Effect

Keshav N. Shrivastava, Senior Member, IACSIT

International Journal of Applied Physics and Mathematics, Vol. 3, No. 3, May 2013

208DOI: 10.7763/IJAPM.2013.V3.207


209

o=ne2/m (3)

where n is the electron density. In the presence of a steady

magnetic field, the conductivity and resistivity become

tensors/matrices,

Fig. 2. To make a two-dimensional electron system, layers of AlGaAs are

deposited over GaAs.

=

yyyx

xyxx

,

=

yyyx

xyxx

(4)

We take x and y axes in the 2-dimensional plane, to obtain,

ix= ,yxyxxx EE

iy= yyxxyy EE

(5)

Owing to the isotropy, xx=yy and xy=-yx. The first one

is called the diagonal conductivity and the second one is

called the Hall conductivity. The relation between

conductivity and resistivity is,

xx=yy= 22

xyxx

xx

(6)

for the diagonal resistivity and for the Hall resistivity, we

obtain,

xy= - yx= - 22

xyxx

xy

(7)

We measure these quantities by connecting various leads

as given in Fig. 3.

Fig. 3. The sample with the magnetic field B and the current I perpendicular

to it.

The wires are connected to measure Hall voltage

perpendicular to both I and B in various directions. W is the

width of the sample and L is the length.

In the case of homogeneous current in the y direction,

ix=I/W and iy=0. The electric field is given by, Ex=V12/L and

Ey=V13/W so we have, xx=V12W/IL and yx= V13/I=RH. The

Hall resistivity in two dimensional electron system is the Hall

resistance per unit area. According to the Drude (ordinary

non-interacting metals) theory,

xx=1/o=m/ne2 and xy=B/nec in a weak magnetic field.

The Hall resistivity is inversely proportional to the electron

density and independent of mean scattering life time. In

strong magnetic fields, there are new phenomena. Von

Klitzing found the plateau in the Hall resistivity which gave

the correct value of h/e2. One of his plateaus is given in Fig. 4.

We see from the plot that (i) there is a plateau region in which

the Hall resistivity remains constant. As the electron density

is varied in this region the diagonal resistivity is almost zero.

(ii) The value of the Hall resistivity in the plateau regions is

exactly equal to h/e2 divided by an integer. Therefore, the

Hall conductivity xy in the plateau region is “quantized” into

integer multiples of e2/h. This phenomenon was called the

“integer quantized Hall effect (IQHE). Since the bending

point occurs a little bit earlier than the plateau, the value is

uncertain by that much.

II. QUANTUM HALL EFFECT

A. Flux Quantization and the Hall Effect.

Introduction of the flux quantization immediately explains

von Klitzing’s plateau. We have the Hall resistivity as,

=B/nec (8)

where B is the magnetic field. According to flux quantization,

the field in a certain area, A, is quantized,

B.A=mo (9)

where the magnetic flux quantum is, o=hc/e. Substitution of

(9) into (8) gives the integer quantized Hall effect as,

=2e

h

N

m

ecA

NAe

mhc

oo

(10)


210

The quantum Hall effect thus is the quantization of Hall

resistivity as,

= 2ie

h. (11)

Fig. 4. The Hall resistivity shows plateau at high fields. The picture shows

von Klitzing’s data at a field of 13 Tesla (13x104Gauss).

Fig. 5. The data of Hall resistivity showing plateau at 1/3 below a

temperature of 1 K. (D.C. Tsui, H.L.Stormer and A.C. Gossard).

Hence the charge of the quasiparticle is ie . Here i=integer.

The charge thus becomes 1e, 2e, 3e, …, ie, … . von Klitzing

obtained the correct value of the charge for i=1. So the value

of h/1e2 became “one von Klitzing”. When this experiment

was repeated with cleaner samples with higher electron

mobility, with higher magnetic fields and lower temperatures,

it lead to the discovery of i=1/3 plateau which gave birth to

the fractional charge. We show the data of Tsui, Stormer and

Gossard in Fig. 5 with plateau at 1/3. Since this fractional

value occurred in the middle of various highly degenerate

Landau levels, where no gap is apparent, the observation

could not be explained by the experimentalists by using the

non-interacting quantum mechanical theory. It was thought

that the observation is a result of the many-body effects of

electron interactions. Subsequent experimental work showed

a lot many more plateaus which are displayed in Fig.6. We

will see that there is no need of interaction to explain the

plateau at 1/3.

Fig. 6. The fractional charges seen symmetrically around ½.

B. Laughlin’s Theory

Laughlin made the first efforts to explain the quantum Hall

effect. The flux quantization was immediately found to

explain at least the integer quantized Hall effect.

Subsequently, Laughlin started from first principles using the

Hamiltonian,

j kj kj

jjjzz

ezVAceiH

||)}()/()/({

22

(12)

where j and k sum over N particles and V is the potential due

to nuclei. The repulsive Coulomb interactions can produce

the plateaus only when flux quantization is considered. A

trial wave function of the form given below,

)(2

1 ),...,( ezz N (13)


211

with =1/m ,

N

kj

kj zmzznm

22 2/2 (14)

and is a multicenter integral, to solve the Schrodinger

equation to find the ground state. Laughlin obtained the

approximate energy expression in terms of m as well as

computed exactly. By using “incompressibility” the charge

of the particles is fixed at 1/3 and 1/5. Unfortunately, there is

area in the flux quantization which must also be fixed,

otherwise the charge will leak. Laughlin also shared the 1998

Nobel prize with Tsui and Stormer but it is now believed that

Laughlin’s wave function is not relevant to explain the data.

C. Shrivastava’s Theory

We consider that electrons have spin as well as the orbital

angular momentum so that,

))((2

1)(

2

1slggjgglgsgjg slsllsj

(15)

Multiplying both sides by j = l +s and taking eigen values,

)]1()1()[((2/1)1()(2/1

)1(

ssllggjjgg

jjg

slsl

j

(16)

Substituting s=1/2 and j = l (1/2) we get,

12

l

gggg ls

lj (17)

For gs=2, gl = 1, we find,

g =

12

11

l . (18)

The equation (17) has both the signs for the spin. The

cyclotron frequency is,

mc

eB (19)

From the charge, e in the cyclotron frequency, we generate

the charge of a particle. It is also possible to obtain the charge

from the e in Bohr magneton. Corresponding to the cyclotron

frequency, the voltage along y direction is,

=eVy. (20)

or,

mc

eB=eVy. Multiplying this expression by e/h, we get,

yVh

e

mc

Be 22

2

(21)

which is the current in x direction, so that the resistivity,

xy= 2e

h (22)

The Sz=1/2, and the energy in a magnetic field is gBH.S,

so correcting B in the cyclotron frequency we find,

Ix=h

Veg

mc

Beg

y

22

2

1

22

1

(23)

For l =0, g=2,

Ix= yVh

e 2

(24)

which describes the quantized current correctly for =1.

From the above equations we have g

2

1

which gives the

filling factor, one for + sign and the other for – sign as in (18).

For l =0, we obtain (1/2)g+=1 and (1/2)g-=0 and other values

as given in Table 1.The Landau levels are introduced by

multiplying the above values by n so that =n(2

1 g) so we

can multiply the tabulated values by an integer when needed.

The values of xx and xy for a single interface of

GaAs/AlGaAs have been measured by Willet et al at 150 mK.

The values predicted in the Table I are exactly the same as in

Willet’s experimental data shown in Fig. 7. The values

shown in the figure occur in two sets,

The values 2/5, 3/7, 4/9, 5/11, 6/13, … etc. and 2/3, 3/5,

4/7, 5/9, 6/11 etc. The predicted values in the table are the

same as in the experimental data. Using the table 1, when we

multiply the values by n we can interpret all of the

experimentally measured values. The columns of Table I

belong to two signs of spin, one belonging to +1/2 and the

other to -1/2. For the cyclotron frequency c=gBB where

B=e/2mc is the Bohr magneton. Therefore, (1/2)g can be

considered to be the effective charge, eeff= ege 2

1 . In Table

I, we see two series, 12

l

l

and 12

1

l

l

which can be

used to explain the high Landau levels easily.

Eisenstein et al have found that for the higher values of the

Landau level quantum number, n, the number of fractions

observed are much less than at the lowest Landau level. At

the magnetic field of 4 or 5 Tesla only a small number of

fractions are observed, the strongest ones are at: 8/3, 5/2 and

7/3. The series )12/( ll is the particle-hole conjugate of

).12/()1( ll For l=7, two values, 7/15 and 8/15 are

predicted and for l= the value is ½. When the same particle

occurs in different levels its charge remains unchanged. We

can multiply the values by n=5 so that the predicted values of

½, 7/15 and 8/15 become 5/2, 7/3 and 8/3. These predicted

values are exactly the same as those observed experimentally

by Eisenstein et al. Thus, 7/3 is the particle-hole conjugate of

8/3 as seen in Table II for n=5.


212

TABLE I: THE PRINCIPAL FRACTIONS OBTAINED FROM THE TWO SERIES

l (1/2)g-=l/(2l+1) (1/2)g+=(l+1)/(2l+1)

0 0 1

1 1/3 2/3

2 2/5 3/5

3 3/7 4/7

4 4/9 5/9

5 5/11 6/11

6 6/13 7/13

1/2 1/2

Fig. 7. The experimental data on Hall resistivity showing fractions of

charges.

Fig. 8. Hall effect data showing plateau at 1/3. The observed series of charges

are exactly the same as in Table I.

For a long time only odd denominators were reported

which show that even denominators are weak. After that even

denominators as well as even numerators with odd

denominators are found. We go back to the same formula,

12

2

1

l

sl . When s=1, l=0, it is 3/2 for + sign. Hence for electron

clusters or pairs the fraction has even denominator. Since the

number of particles is larger than one its probability becomes

small so these plateaus are weak but the same theory explains

the even denominators. There is a limiting value of the series

which also gives ½. One can introduce a Fermi surface at n/2,

Thus ½, 2/2, 3/2, 4/2, 5/2, 6/2 and 7/2 are predicted which are

observed by Yeh et al. [8] as shown in Fig. 9.

Fig. 9. The predicted n/2 is observed in this experimental data.

TABLE II: EFFECT OF N THE LANDAU LEVEL NUMBER

l l/(2l+1) (l+1)/(2l+1) nl/(2l+1) n(l+1)/(2l+1)

1/2 1/2 5/2 5/2

7 7/15 8/15 7/3 8/3

It has been reported by Yeh et al. [8] that the effective

mass and g factor of some of the fractions are equal to those

of others. We have shown that the effective mass can be

equal only when the two quasiparticles are particle-hole

conjugates. The particle-hole conjugates should obey the

following relation,

p+h=1 (25)

The values given in Table I always obey this relation.

D. Half filled Landau Level

The l = in the two series produces,

)(2

1

)(2

1

lim

lim

l

l (26)

One ½ comes from the right and the other from the left

when magnetic field is varied, i.e., one while increasing the

field and the other while reducing. One can go from 1/3 to ½

by reducing the field while from 2/3 to ½ is obtained by

increasing the field. We can go from 1/3 to 3/5 by reversing

the spin and increasing the l. Similarly from 2/3 to 2/5 by

reducing the spin and increasing l. In this way angular

momentum is conserved. One of the ½ values is like an

electron (1/2)A and the other is like a hole, (1/2)B, (A for +

series and B for – series). Since the electron and the hole are

separated by a distance, this state is compressible. When we


213

multiply this result by n, the Landau level quantum number,

we obtain: ½, 2/2, 3/2, 4/2, 5/2, …, which are in agreement

with data.

E. Effective Charge

We discussed the Laughlin’s wave function earlier. Here

the repulsive Coulomb interactions can not give rise to a

fractional charge of 1/3. It is first assumed on the basis of

experimental data and then substituted in the theory.

Laughlin’s charge is independent of spin but in our theory it

depends. In Laughlin’s theory a particle of charge 1/3 is

produced but in our theory, splitting occurs in fractions of 1/3

and 2/3, etc.

Laughlin’s 1/3 charge.

Shrivastava’s theory.

(1/3)e+(2/3)e =1e

(2/5)e+(3/5)e=1e, etc.

Fig. 10. There is a difference between Laughlin and our

theories. The fractionalization in Laughlin’s theory is

independent of spin. Whereas in our theory spin plays an

important role in determining the fraction.

F. Dirac Equation

The basic idea of Dirac equation is to have space-time

symmetry and the constancy of velocity of light, so that

instead of p2/2m, the kinetic energy appears as c.p and the

wave equation becomes,

),(),().( 2 txt

itxmcpc

(27)

The free particle solutions of which are,

2/14222 )( cmpcE . (28)

This equation gives the correct magnetic moments for the

proton as well as for the neutron subject to using the mass of

the respective particle and an appropriate g value. In the case

of electron Lande’s formula,

)1(2

)1()1()1(1

jj

sslljjg (29)

with positive spin is used. In our case, the effective charge

and hence the magnetic moment is determined by the g

values. Hence, our method of defining the charge of the

electron is the same as that for the magnetic moments of

proton and neutron.

G. Shubnikov-de Haas Effect

At low temperatures, the integration over the Fermi

distributon leads to x/sinh x

Type expression which is called Dingle’s formula. The

spin symmetry is found to modify this formula which

determines the oscillation amplitude of resistivity as a

function of magnetic field, called the Shubnikov-de Haas

effect. Our theory introduces the effective charge so that the

cyclotron frequency gets fractionalized resulting into m/

which for =1 becomes m, the electron mass. Thus, we have

taken into account, the spin-charge fractions, to obtain the

correct mass. For example, at certain magnetic field 1.5 m is

found instead of m. The mass of the electron relative to band

value as a function of carrier density deduced from

Shubnikov-de Haas effect in GaAs/AlGaAs heterostructures

is shown in the Fig. 11. The factors 1 and 2/3 are found to

arise from the spin-charge effect of Shrivastava. The

experimental data is obtained from Tan, Stormer et al. [9 ].

The dashed line is found by Kwon et al on the basis of small

self energy corrections due to many body perturbative

interactions. The factors of 1 and 2/3 in the mass are found by

us. The mass of the free electron is me and the screening

radius is deduced from the density, n, per unit area. We find

that m/ occurs in place of m for the mass of the electron in

the Shubnikov-de Haas (SdH) effect. In fact, many other

fractions of the mass of the electron given in Table 1, become

allowed so that the electron really “falls apart”. The

oscillations due to flux quantization allow the measurement

of m/h2. The flux quantization in the Shubnikov-de Haas

effect leads to “quantized Shubnikov-de Haas effect”.

Therefore, we observe consequences of the effect of flux

quantization on the Shubnikov-de Haas oscillations. There

are zeroes in the resistivity at certain fields. There is a

spin-charge effect so that the spin flip corresponds to a

change in the charge.

The Shubnikov-de Haas effect uses quantization of

Landau levels but not the flux quantization. Hence, we find

that there is a “quantized Shubnikov-de Haas effect” which

measures the m/h2. We find that when fractional values of

are taken into account, the mass of the electron, equal to band

mass in GaAs/AlGaAs is obtained. When magnetic field is

varied, the different values of n cross the Fermi energy at

different fields resulting into oscillations in the resistivity as a

function of magnetic field. The oscillating resistivity is given

by, resistivity is given by,

)]2

1(2cos[)(

/2sinh

/2

)(exp

2

2

,

c

cB

cBth

qc

thpoxx

pc

vTpk

vTpk

where

cv

pc

(30)


214

The cosine factor also leads to zero resistivity when ever,

22

12

cvp

(31)

so that the resistivity vanishes when B satisfies the above

formula. We introduce the flux quantization so that the

exponential factor in (30) becomes,

hv

mA

eBv

mcP

cc

expexp (32)

so that m/h will be measured from the oscillations.

Introducing the flux quantization in the argument of Sinh

factor, we obtain,

mA

hnc

2

(33)

which does not have the charge but measures m/h. In the

experiments the factor measured is m*g*/n so it is clear that

the mass and g get mixed.

H. Spin-Charge Locking

The charge of the electron may be described by matrices

just as the angular momentum is. When the spin is aligned

along the charge, such as sx parallel to ex, the arrangement is

called the spin-charge locking. Taking our effective charge

expression for eeff/e=(1/2)g we find the dot product of spin

and charge to find,

]'.').1()'.[(12

1

.2

112. ,1,

ssssll

ssllse

(34)

which produces spin-orbit and spin-spin interactions and

there is spin divided by 2l+1. That is what makes it difficult

to detect this type of effect.

We have

thus found the correct explanation of the

experimentally observed quantum Hall effect. We find that

angular momentum gives rise to fractional charge. Therefore,

there is a spin-charge effect, i.e., under high magnetic fields,

the spin determines the charge the Lande’s formula is

replaced by a linear formula and Landau levels are modified

[11-24].

Laughlin’s wave function is not the interpretation of

quantum Hall effect [25].

III.

HELICITY IN QUANTUM HALL EFFECT

We look at the charge of the electrons in the quantum Hall

effect as

e*=(1/2)g±e.

For l

= 0,

see 2

1/*

(35)

Hence we predict particles of charge 0 and 1. The particle

of charge zero uses the negative sign and hence its helicity is

negative. The particle of charge 1 uses the positive sign.

Hence its helicity is positive. The particles occur in pairs.

Hence particle is defined by positive helicity and its charge is

equal to that of the electron charge. The particle of charge

zero has negative helicity and hence it is the antiparticle

analog of the electron of charge 1.

Particle: helicity positive, charge =1,

Antiparticle: helicity negative, charge =0.

For l =1, s=1/2, e*=1/3 with negative helicity and e*=2/3

for positive helicity. These values are exactly the same as

observed in the experimental data. Hence we can determine

the helicities of all of the plateaus. The particle of charge 1/3

is left handed. Li et al. [10] have performed the experimental

measurements in AlGaAs from which we observe that the

plateaus occur at, 1/5, 2/9, 2/3, 3/5, 2/5, 1/3, 5/3, 4/3, 4/7,

7/11, 3/7 and 63/100.

We explain all of these values and determine the helicity in

each case.

1) For 2l+1=5, l =2, s=1/2, the principal fractions occur at

l/(2l+1) and ( l +1)/(2l+1) so that 2/5 has negative

helicity and 3/5 has positive helicity. The resonance

occurs at 3/5-2/5=1/5. This explains the plateau at 1/5 as

a “resonance”. It belongs to mixed helicity. It can rotate

and the velocity can change sign during rotation such

that 3/5 can become 2/5 and 2/5 can become 3/5 so that

3/5+2/5=2/5+3/5 =1 becomes a particle of charge 1 with

mixed helicity and 3/5-2/5 =1/5 becomes 2/5-3/5=-1/5

which is a hole in the band theory. The hole has the

charge of opposite sign compared with that of the

electron. Thus we have particles of mixed helicities with

charge ±1/5 as well as 2/5 and 3/5.

2) The charge 2/9 has the denominator of 2l+1=9. Hence, l

=4. The l /(2l+1)=4/9 and (l +1)/(2l+1) = 5/9. The

resonance occurs at 5/9-4/9=1/9 and 4/9 has negative

helicity whereas 5/9 has positive helicity. The charge 1/9

has mixed helicity and the two particle state at

1/9+1/9=2/9 has mixed helicity.

3) The state 2/3 has l =1 and (l +1)/(2l+1)=2/3 has positive

helicity and l/(2l+1)=1/3 has negative helicity.

4) The states 2/5 and 3/5 are the l =2, s=1/2 with

l/(2l+1)=2/5 for negative helicity and (l +1)/(2l+1)=3/5

for positive helicity.

5) The state 5/3>1 is not due to the principal fractions. It

requires that a contribution should come from the

bosonic character of the Landau levels. It is possible to

make 5/3 as 1/3+1/3+1/3+1/3+1/3=5/3 which is of

negative helicity. It is also possible to make

2/3+2/3+1/3=5/3 which is of mixed helicity. It can rotate

to make 2/3+1/3+2/3=5/3 which is an example of

“reversed helicity”. It is possible to make a state

1/3+1/3+2/3=4/3 which is a case of mixed helicity. If

2/3+2/3+1/3=5/3 is a particle then 1/3+1/3+2/3=4/3 is

the antiparticle analog of 5/3. The 4/3 as well as 5/3 are

degenerate because there is more than one way of

making them. The state 2/3+2/3=4/3 as well as

1/3+1/3+2/3=4/3 are degenerate.

6) The denominator 2l+1=7 is made from l = 3. Hence, the

principal value for negative helicity l /(2l+1)=3/7 and


215

for positive helicity, ( l +1)/(2l+1)= 4/7.

7) For 2l+1=11, l =5. Hence the principal value for positive

helicity is 6/11 and for negative helicity it is 5/11. The

resonance occurs at 6/11-5/11=1/11. The sum process

gives 6/11+1/11 =7/11 which is of mixed helicity.

8) The value of 63/100 can be written as

(1/2)(63/50)=(1/2)(1/2)(63/25). For a large l =12, 2

l+1=25. Hence 12/25 and 13/25 are the

particle-antiparticle conjugates. The 63/25 is the sum as

50/25+13/25=2+13/25 which is above the second

Landau level. The flux quantization occurs at n’hc/e. For

n’=2 the charge reduces to ½ as hc/(e/2) so that we can

reduce the charge to ½ of its value. Hence 63/25

becomes 63/50. For l +(1/2)±s=63, l =12 we have

±s=63-12-(1/2)=101/2 which is possible in a cluster of

electrons. For Landau levels, for g1=0, n2=0,

2224

1)

2

1(

2

1gng

(36)

Hence a factor of ¼ comes so that 63/25 becomes 63/100.

The Landau levels with suitable quantum numbers with

positive helicity explain the value of 63/100. In this way, the

“principal fractions”, the resonances, the sum process as well

as clustering explain the experimental data of AlGaAs.

9) For s=0, (1/2)g=1/2 so that e*/e =1/2 for both helicities.

For s=0, the concept of particle and antiparticle

disappears and the particles are said to become

“Majorana type”.

IV. BREMSSTRAHLUNG

Bremsen means to brake (slow down) and Strahlung

means radiation, i.e., radiation produced by slowing down

the electrons. The electron with spin ½ and energy E2 slows

down to a lower energy E1 and the energy is conserved by

emission of radiation. There is an electron of energy E2 in the

initial state in the initial state and there is an electron with

energy E1 as well as a photon in the final state, E2=E1+h. In

the quantum Hall effect, a particle of charge 1/3 travelling

with negative helicity (left handed) emits a photon

(unpolarized) and a particle of charge 1/3 continues to

propagate. It is possible that a particle of charge 1/3 emits a

particle of zero charge and a particle of 1/3 charge. A particle

of charge 2/3 moving with positive helicity can slow down to

emit two particles one of charge 2/3 and the other of charge

zero.

V. CONCLUSION

The quantum Hall effect first observed by von Klitzing et

al had no theory at all and the origin of fractions has not been

explained by them. In particular they have not explained the

fractional charge at which the plateaux occur. We have

explained the fractions from our original theory [7] and found

that Lande’s as well as Landau’s theories require

modifications. The role of angular momentum in the

understanding of the quantum Hall effect is quite clearly

demonstrated.

Fig. 11. The mass of the electron deduced from the amplitude of oscillations

as affected by our charge factors.

REFERENCES

[1] E. Hall, “On a new action of the magnet on electric currents,” American

J. Maths, vol. 2, pp. 287-292, 1879.

[2] T. Englert and K. V. Klitzing, Surf. Sci., vol. 73, no. 70, 1978.

[3] K. V. Klitzing, G. Dorda, and M. Pepper, “New method for

high-accuracy determination of the fine structure constant,” Phys. Rev.

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[4] D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional

magnetotransport in the extreme quantum limit,” Phys. Rev. Lett., vol.

48, pp. 1559-1562, 1982.

[5] R. Willett, J. P. Eisenstein, H. L. Stormer, D. C. Tsui, A. C. Gossard,

and J. H. English, “Observation of an even-denominator quantum

number in the fractional quantum Hall effect,” Phys. Rev. Lett., vol. 59,

pp. 1776-1779, 1987.

[6] R. B. Laughlin, “Anomalous quantum Hall effect: An incompressible

quantum fluid with fractionally charged excitations,” Phys. Rev. Lett.,

vol. 50, pp. 1395-1398, 1983.

[7] K. N. Shrivastava, “Rational numbers of the fractionally quantized Hall

effect,” Phys. Lett. A, vol. 113, pp. 435-436, 1986.

[8] A. S. Yeh, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, K.

W. West, Phys. Rev. Lett., vol. 62, pp. 592, 1999.

[9] Y. W. Tan, J. Zhu, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and K.

W. West, “Measurement of the density-dependent many-body electron

mass in two dimensional GaAs/AlGaAs heterostructures,” Phys. Rev.

Lett., vol. 94, pp. 016405-1-4, 2005.

[10] W. Li et al., Phys. Rev. Lett., vol. 105, pp. 076803, 2010.

[11] K. N. Shrivastava, “Negative-spin quasiparticles in quantum Hall

effect,” Phys. Lett. A, vol. 326, pp. 469-472, 2004.

[12] K. N. Shrivastava, “Particle-hole symmetry in quantum Hall effect,”

Mod. Phys. Lett. B, vol. 13, pp. 1087-1090, 1999.

[13] K. N. Shrivastava, “Theory of quantum Hall effect with high Landau

levels,” Mod. Phys. Lett. B, vol. 14, pp. 1009-1013, 2000.

[14] K. N. Shrivastava, Frontiers of fundamental physics 4, 1st ed. B. G.

Sidharth, N.Y.: Academic/Plenum Pub., 2001, pp.235-249.

[15] K. N. Shrivastava, Introduction to quantum Hall effect, New York:

Nova Sci. Pub., 2002.

[16] K. N. Shrivastava, Quantum Hall effect: Expressions, New York: Nova

Science Pub., 2005.

[17] K. N. Shrivastava, “Additional Dirac matrix in quantum Hall effect,” in

Proc. AIP, vol. 1017, pp. 47-56, 2008.

[18] K. N. Shrivastava, “The theory of the quantum Hall effect,” in Proc.

AIP, vol. 1017, pp. 422-428, 2008.

[19] N. A. Zabidi, H. A. Kassim, and K. N. Shrivastava, “DFT calculation of

band structure of carbon chain pulled from grapheme,” in Proc. AIP,

vol. 1250, pp. 241-244, 2010.


216

[20] N. A. Rosli, H. A. Kassim, and K. N. Shrivastava, “Graphene infrared

spectroscopy: DFT vibrational frequencies,” in Proc. AIP, vol. 1250,

pp. 269-272, 2010.

[21] V. R. Devi and K. N. Shrivastava, “DFT calculation of vibrational

frequencies of GaAs and the Raman spectra,” Spectrochimica Acta A,

vol. 95, pp. 172-176, 2012.

[22] K. N. Shrivastava, “The Lande’s g value in quantum dots in AlGaAs,”

International Journal of Nanoscience, World Sci., vol. 10, pp. 507-514,

2011.

[23] K. N. Shrivastava, “The quantum Hall effect,” in Proc. AIP, vol. 1482,

pp. 335-339, 2012.

[24] M. M. A. Ali and K. N. Shrivastava, “Symmetry properties of

Laughlin’s wave function and related states,” in Proc. AIP, vol. 1482,

pp. 43-46, 2012.

[25] K. N. Shrivastava, “Laughlin’s wave function and angular

momentum,” International Journal of Modern Physics B, vol. 25, pp.

1301-1357, 2011.

Keshav N. Shrivastava was born on July 11, 1943.

He obtained his B.Sc. degree from the Agra

University in 1961, M.Sc. degree from the University

of Allahabad in 1963, Ph.D. from the Indian Institute

of Technology Kanpur in 1966 and D.Sc. from

Calcutta University in 1980.

He worked at the Clark University during 1966-68,

at the Harvard University during the summer of 1968, at the Montana State

University during 1968-69, and at the University of Nottingham during

1969-70. He was Associate Professor at the Himachal Pradesh University

Simla, India during 1970-74. He worked at the University of California Santa

Barbara during 1974-75 and at the State University Utrecht during 1975-76.

He was Professor at the University of Hyderabad where he worked from

1978 till 2005. He worked in Tohoku University during 1999-2000. He

visited the University of Zurich for short times during 1984, 1985, 1988 and

1999. He worked in the University of Houston in the summer of 1990. He

visited the Royal Institute of Technology Stockholm and the University of

Uppsala during 1994. He visited the Technical University of Vienna, Austria

in the summer of 1995. He visited the Princeton University and the

University of Cincinnati during 2003. He was Professor in the University of

Malaya Kuala Lumpur during 2005-2011. He is affiliated with the University

of Hyderabad, India. He has worked in electron spin resonance, spin-phonon

interaction, relaxation, Moessbauer effect, exchange interaction,

superconductivity and the quantum Hall effect. He discovered the flux

quantized energy levels in superconductors, the spin dependent flux and the

correct theory of the quantum Hall effect. He has published more than 227

papers in journals and he is the author of the following books. (i)

Superconductivity: Elementary Topics, World Scientific Singapore 2000. (ii)

Introduction to Quantum Hall effect, Nova Sci. Pub N.Y. 2002 and (iii) The

quantum Hall effect: Expressions, Nova Sci. N.Y. 2005.

Prof. Shrivastava is a Fellow of the Institute of Physics London, Fellow of

the National Academy of Sciences India and member of the American

Physical Society.

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